7.49 Sheafification in a topology
In this section we explain the analogue of the sheafification construction in a topology.
Let \mathcal{C} be a category. Let J be a topology on \mathcal{C}. Let \mathcal{F} be a presheaf of sets. For every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) we define
L\mathcal{F}(U) = \mathop{\mathrm{colim}}\nolimits _{S \in J(U)^{opp}} \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})
as a colimit. Here we think of J(U) as a partially ordered set, ordered by inclusion, see Lemma 7.47.2. The transition maps in the system are defined as follows. If S \subset S' are in J(U), then S \to S' is a morphism of presheaves. Hence there is a natural restriction mapping
\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S', \mathcal{F}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}).
Thus we see that S \mapsto \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) is a directed system as in Categories, Definition 4.21.2 provided we reverse the ordering on J(U) (which is what the superscript {}^{opp} is supposed to indicate). In particular, since h_ U \in J(U) there is a canonical map
\ell : \mathcal{F}(U) \longrightarrow L\mathcal{F}(U)
coming from the identification \mathcal{F}(U) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}). In addition, the colimit defining L\mathcal{F}(U) is directed since for any pair of covering sieves S, S' on U the sieve S \cap S' is a covering sieve too, see Lemma 7.47.2.
Let f : V \to U be a morphism in \mathcal{C}. Let S \in J(U). There is a commutative diagram
\xymatrix{ S \times _ U V \ar[r] \ar[d] & h_ V \ar[d] \\ S \ar[r] & h_ U }
We can use the left vertical map to get canonical restriction maps
\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) \to \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S \times _ U V, \mathcal{F}).
Base change S \mapsto S \times _ U V induces an order preserving map J(U) \to J(V). And the restriction maps define a transformation of functors as in Categories, Lemma categories-lemma-functorial-colimit. Hence we get a natural restriction map
L\mathcal{F}(U) \longrightarrow L\mathcal{F}(V).
Lemma 7.49.1. In the situation above.
The assignment U \mapsto L\mathcal{F}(U) combined with the restriction mappings defined above is a presheaf.
The maps \ell glue to give a morphism of presheaves \ell : \mathcal{F} \to L\mathcal{F}.
The rule \mathcal{F} \mapsto (\mathcal{F} \xrightarrow {\ell } L\mathcal{F}) is a functor.
If \mathcal{F} is a subpresheaf of \mathcal{G}, then L\mathcal{F} is a subpresheaf of L\mathcal{G}.
The map \ell : \mathcal{F} \to L\mathcal{F} has the following property: For every section s \in L\mathcal{F}(U) there exists a covering sieve S on U and an element \varphi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) such that \ell (\varphi ) equals the restriction of s to S.
Proof.
Omitted.
\square
Definition 7.49.2. Let \mathcal{C} be a category. Let J be a topology on \mathcal{C}. We say that a presheaf of sets \mathcal{F} is separated if for every object U and every covering sieve S on U the canonical map \mathcal{F}(U) \to \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) is injective.
Theorem 7.49.3. Let \mathcal{C} be a category. Let J be a topology on \mathcal{C}. Let \mathcal{F} be a presheaf of sets.
The presheaf L\mathcal{F} is separated.
If \mathcal{F} is separated, then L\mathcal{F} is a sheaf and the map of presheaves \mathcal{F} \to L\mathcal{F} is injective.
If \mathcal{F} is a sheaf, then \mathcal{F} \to L\mathcal{F} is an isomorphism.
The presheaf LL\mathcal{F} is always a sheaf.
Proof.
Part (3) is trivial from the definition of L and the definition of a sheaf (Definition 7.47.10). Part (4) follows formally from the others.
We sketch the proof of (1). Suppose S is a covering sieve of the object U. Suppose that \varphi _ i \in L\mathcal{F}(U), i = 1, 2 map to the same element in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, L\mathcal{F}). We may find a single covering sieve S' on U such that both \varphi _ i are represented by elements \varphi _ i \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S', \mathcal{F}). We may assume that S' = S by replacing both S and S' by S' \cap S which is also a covering sieve, see Lemma 7.47.2. Suppose V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), and \alpha : V \to U in S(V). Then we have S \times _ U V = h_ V, see Lemma 7.47.5. Thus the restrictions of \varphi _ i via V \to U correspond to sections s_{i, V, \alpha } of \mathcal{F} over V. The assumption is that there exist a covering sieve S_{V, \alpha } of V such that s_{i, V, \alpha } restrict to the same element of \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S_{V, \alpha }, \mathcal{F}). Consider the sieve S'' on U defined by the rule
7.49.3.1
\begin{eqnarray} \label{sites-equation-S-prime-prime} (f : T \to U) \in S''(T) & \Leftrightarrow & \exists \ V , \ \alpha : V \to U, \ \alpha \in S(V), \nonumber \\ & & \exists \ g : T \to V, \ g \in S_{V, \alpha }(T), \\ & & f = \alpha \circ g \nonumber \end{eqnarray}
By axiom (2) of a topology we see that S'' is a covering sieve on U. By construction we see that \varphi _1 and \varphi _2 restrict to the same element of \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S'', L\mathcal{F}) as desired.
We sketch the proof of (2). Assume that \mathcal{F} is a separated presheaf of sets on \mathcal{C} with respect to the topology J. Let S be a covering sieve of the object U of \mathcal{C}. Suppose that \varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(S, L\mathcal{F}). We have to find an element s \in L\mathcal{F}(U) restricting to \varphi . Suppose V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), and \alpha : V \to U in S(V). The value \varphi (\alpha ) \in L\mathcal{F}(V) is given by a covering sieve S_{V, \alpha } of V and a morphism of presheaves \varphi _{V, \alpha } : S_{V, \alpha } \to \mathcal{F}. As in the proof above, define a covering sieve S'' on U by Equation (7.49.3.1). We define
\varphi '' : S'' \longrightarrow \mathcal{F}
by the following simple rule: For every f : T \to U, f \in S''(T) choose V, \alpha , g as in Equation (7.49.3.1). Then set
\varphi ''(f) = \varphi _{V, \alpha }(g).
We claim this is independent of the choice of V, \alpha , g. Consider a second such choice V', \alpha ', g'. The restrictions of \varphi _{V, \alpha } and \varphi _{V', \alpha '} to the intersection of the following covering sieves on T
(S_{V, \alpha } \times _{V, g} T) \cap (S_{V', \alpha '} \times _{V', g'} T)
agree. Namely, these restrictions both correspond to the restriction of \varphi to T (via f) and the desired equality follows because \mathcal{F} is separated. Denote the common restriction \psi . The independence of choice follows because \varphi _{V, \alpha }(g) = \psi (\text{id}_ T) = \varphi _{V', \alpha '}(g'). OK, so now \varphi '' gives an element s \in L\mathcal{F}(U). We leave it to the reader to check that s restricts to \varphi .
\square
Definition 7.49.4. Let \mathcal{C} be a category endowed with a topology J. Let \mathcal{F} be a presheaf of sets on \mathcal{C}. The sheaf \mathcal{F}^\# := LL\mathcal{F} together with the canonical map \mathcal{F} \to \mathcal{F}^\# is called the sheaf associated to \mathcal{F}.
Proposition 7.49.5. Let \mathcal{C} be a category endowed with a topology. Let \mathcal{F} be a presheaf of sets on \mathcal{C}. The canonical map \mathcal{F} \to \mathcal{F}^\# has the following universal property: For any map \mathcal{F} \to \mathcal{G}, where \mathcal{G} is a sheaf of sets, there is a unique map \mathcal{F}^\# \to \mathcal{G} such that \mathcal{F} \to \mathcal{F}^\# \to \mathcal{G} equals the given map.
Proof.
Same as the proof of Proposition 7.10.12.
\square
Comments (2)
Comment #5861 by Tathagata Basak on
Comment #6072 by Johan on